Properties

Label 7728.2.a.bu
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} - q^{7} + q^{9} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{11} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} ) q^{15} + \beta_{2} q^{17} + ( -2 + \beta_{2} ) q^{19} - q^{21} - q^{23} + ( 8 - \beta_{1} ) q^{25} + q^{27} + ( -2 - 3 \beta_{2} ) q^{29} + ( -2 \beta_{1} + \beta_{2} ) q^{31} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} ) q^{35} + ( 2 \beta_{1} - \beta_{2} ) q^{37} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( 3 - 3 \beta_{1} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + \beta_{2} q^{51} + ( -3 + \beta_{1} + \beta_{2} ) q^{53} + ( 1 + 3 \beta_{1} + 7 \beta_{2} ) q^{55} + ( -2 + \beta_{2} ) q^{57} + ( 9 - \beta_{1} - \beta_{2} ) q^{59} + ( -11 + \beta_{1} ) q^{61} - q^{63} + ( -6 + 10 \beta_{1} + \beta_{2} ) q^{65} + ( -5 + \beta_{1} + \beta_{2} ) q^{67} - q^{69} + ( -1 + 5 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 6 - 4 \beta_{1} - \beta_{2} ) q^{73} + ( 8 - \beta_{1} ) q^{75} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{77} + ( -4 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{83} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{85} + ( -2 - 3 \beta_{2} ) q^{87} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{89} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{91} + ( -2 \beta_{1} + \beta_{2} ) q^{93} + ( 9 - \beta_{1} - 3 \beta_{2} ) q^{95} + ( 2 + 2 \beta_{1} + 5 \beta_{2} ) q^{97} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + 6q^{11} - 9q^{13} - 3q^{15} - 6q^{19} - 3q^{21} - 3q^{23} + 24q^{25} + 3q^{27} - 6q^{29} + 6q^{33} + 3q^{35} - 9q^{39} + 12q^{41} + 9q^{43} - 3q^{45} + 3q^{49} - 9q^{53} + 3q^{55} - 6q^{57} + 27q^{59} - 33q^{61} - 3q^{63} - 18q^{65} - 15q^{67} - 3q^{69} - 3q^{71} + 18q^{73} + 24q^{75} - 6q^{77} + 3q^{81} + 12q^{83} + 21q^{85} - 6q^{87} + 3q^{89} + 9q^{91} + 27q^{95} + 6q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.523976
2.66908
−2.14510
0 1.00000 0 −3.67750 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −3.21417 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 3.89167 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7728.2.a.bu 3
4.b odd 2 1 3864.2.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.k 3 4.b odd 2 1
7728.2.a.bu 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} + 3 T_{5}^{2} - 15 T_{5} - 46 \)
\( T_{11}^{3} - 6 T_{11}^{2} - 15 T_{11} + 84 \)
\( T_{13}^{3} + 9 T_{13}^{2} - 9 T_{13} - 164 \)
\( T_{17}^{3} - 9 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -46 - 15 T + 3 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 84 - 15 T - 6 T^{2} + T^{3} \)
$13$ \( -164 - 9 T + 9 T^{2} + T^{3} \)
$17$ \( 4 - 9 T + T^{3} \)
$19$ \( -6 + 3 T + 6 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -262 - 69 T + 6 T^{2} + T^{3} \)
$31$ \( -74 - 39 T + T^{3} \)
$37$ \( 74 - 39 T + T^{3} \)
$41$ \( 18 + 9 T - 12 T^{2} + T^{3} \)
$43$ \( 216 - 27 T - 9 T^{2} + T^{3} \)
$47$ \( -232 - 72 T + T^{3} \)
$53$ \( -2 + 15 T + 9 T^{2} + T^{3} \)
$59$ \( -628 + 231 T - 27 T^{2} + T^{3} \)
$61$ \( 1262 + 357 T + 33 T^{2} + T^{3} \)
$67$ \( 72 + 63 T + 15 T^{2} + T^{3} \)
$71$ \( -828 - 153 T + 3 T^{2} + T^{3} \)
$73$ \( 686 + 15 T - 18 T^{2} + T^{3} \)
$79$ \( 344 - 93 T + T^{3} \)
$83$ \( 82 + 21 T - 12 T^{2} + T^{3} \)
$89$ \( 24 - 159 T - 3 T^{2} + T^{3} \)
$97$ \( 1656 - 207 T - 6 T^{2} + T^{3} \)
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